A hybrid adaptive method for initial-boundary value problems

It is well-known that higher-order methods (as compared to lower order accurate methods) capture transient phenomena more efficiently since they allow for a considerable reduction in the degrees of freedom for a given error tolerance. In particular, high-order finite difference methods (HOFDMs) are ideally suited for problems of this type, cf. the pioneering paper by Kreiss and Oliger [5]. For long-time simulations, it is imperative to use finite difference approximations that do not allow growth in time if the PDE does not allow growth—a property termed time stability [3]. Achieving time-stable HOFDM has received considerable past attention. A robust and well-proven high-order finite difference methodology, for well-posed initial boundary value problems (IBVP), is to combine summation-by-parts (SBP) operators [4, 6] and either the simultaneous approximation term (SAT) method [1], or the projection method [7] to impose boundary conditions.